Fast circulant block Jacket transform based on the Pauli matrices

Abstract

Owing to its orthogonality, simplicity of the inversion and fast algorithms, Jacket transform generalising from the Hadamard transform has played important roles in signal and image processing, mobile communication for coding design, cryptography, etc. In this paper, inspired by the emerging block Jacket transform, a new class of circulant block Jacket matrices (CBJMs) are mathematically defined based on the circulant matrix theory. Then the existence conditions for the CBJMs with any size based on the Pauli matrices are explicitly given. Next, by employing the Kroneker product and successive low order basic matrices, the fast algorithms for the construction and decomposition of any high order circulant Pauli block Jacket matrices (CPBJMs) are systematically obtained. Finally, compared to the direct computation (DC), the proposed fast algorithms have a better efficiency, which may be available in many fields, such as signal sequence design, image compression, communication for coding and encoding, quantum signal processing and information theory.